# Square-integrability of solutions of the Yamabe equation

by

Bernd Ammann, Mattias Dahl, Emmanuel Humbert

**Square-integrability of solutions of the Yamabe equation**
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*Commun. Anal. Geom.*
**21**, *891-916* (2013)

http://dx.doi.org/10.4310/CAG.2013.v21.n5.a2

We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^{p} for p=2n/(n-2) are also L^{2}. This L^{p}-L^{2}-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As an application we see that the smooth Yamabe invariant of any 2-connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions ≥ 11.

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The Paper was written on 11.11.2011

Last update 9.1.2014